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RATE OF DECOHERENCE FOR AN ELECTRON WEAKLY COUPLED TO A PHONON GAS

Riccardo Adami, László Erdős

Published Nov 20, 2012• 5 citations
OpenAlex

Abstract

Abstract. We study the dynamics of an electron weakly coupled to a phonon gas. The initial state of the electron is the superposition of two spatially localized distant bumps moving towards each other, and the phonons are in a thermal state. We investigate the dynamics of the system in the kinetic regime and show that the time evolution makes the non-diagonal terms of the density matrix of the electron decay, destroying the interference between the two bumps. We show that such a damping effect is exponential in time, and the related decay rate is proportional to the total scattering cross section of the electron-phonon interaction. 1. Introduction and

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Abstract. We study the dynamics of an electron weakly coupled to a phonon gas. The initial state of the electron is the superposition of two spatially localized distant bumps moving towards each other, and the phonons are in a thermal state. We investigate the dynamics of the system in the kinetic regime and show that the time evolution makes the non-diagonal terms of the density matrix of the electron decay, destroying the interference between the two bumps. We show that such a damping effect is exponential in time, and the related decay rate is proportional to the total scattering cross section of the electron-phonon interaction. 1. Introduction and
To lowest order in the electron-phonon coupling causing electron transitions among electronic eigenstates, the extent (on the time axis) of the memory functions wrs(t) governing the state population via the generalized master equations for the diagonal elements of the electron density matrix ρ(t) is independent of the coupling, being given by the decay time of the bath correlation functions. However, a typical reciprocal decay time of some higher-order contributions to wrs(t) has been found to be given by the reciprocal decay time of the off-diagonal elements of ρ(t), i.e. to be proportional to the square of the electron-phonon coupling constant. This feature makes such weakly damped contributions non-negligible even in lowest-order dynamics, as is illustrated on a dimer.
We theoretically investigate the nonequilibrium relaxation of a spatial density modulation in a one-dimensional, weakly interacting Bose gas, and its connection to the equilibrium scattering rate γk∝k3/2\smash{γ_k\propto k^{3/2}}γk​∝k3/2 of the system's phononic excitations. We show that the relaxation is generally governed by a nonequilibrium scattering rate γk,tγ_{k,t}γk,t​ coupled to quantum fluctuations, which approaches its equilibrium value γkγ_kγk​ only at long times. Numerical simulations of quantum kinetic equations reveal an algebraic convergence, γk,t−γk∼t−2/3\smash{γ_{k,t} - γ_k \sim t^{-2/3}}γk,t​−γk​∼t−2/3, confirmed by analytical predictions. More broadly, our results establish a theoretical framework for experimentally probing phonon dynamics through the temporal evolution of local perturbations in quantum gases.
We study electron-phonon interaction induced decoherence between two-electron singlet and triplet states in a semiconductor double quantum dot using a spin-boson model. We investigate the onset and time evolution of this dephasing, and study its dependence on quantum dot parameters such as dot size and double dot separations, as well as the host materials (GaAs and Si). We find that electron-phonon interaction causes an initial Gaussian decay of the off-diagonal density matrix element in the singlet-triplet Hilbert space, and a long-time exponential decay originating from phonon relaxation.
The exact quantum mechanics of a model semiconductor system of an electron with a discrete and equidistant energy spectrum interacting with a single phonon mode is presented. An electron initially excited into a coherent superposition of states interacts with phonons thereby creating a self-interference in time which reduces the emission of phonons to isolated bursts. This self-interference effect gives rise to steplike behavior in the relaxation kinetics of the electron and phonons. We show that this ``phonon staircase'' effect is a consequence of a correlated initial electron distribution and the violation of energy conservation in the electron-phonon interaction on short time scales.
It is generally accepted that the phonon gas of a superfluid always enters a weak coupling regime at sufficiently low temperatures, whatever the strength of the interactions between the underlying particles (constitutive of the superfluid). Thus, in this limit, we should always be able to calculate the damping rate of thermal phonons by applying Fermi's golden rule to the Hamiltonian H3H_3H3​ of cubic phonon-phonon coupling taken from quantum hydrodynamics, at least in the case of a convex acoustic branch and in the collisionless regime (where the eigenfrequency of the considered phonons remains much greater than the gas thermalization rate). Using the many-body Green's function method, we predict that, unexpectedly, this is not true in two dimensions, contrary to the three-dimensional case. We confirm this prediction with classical phonon-field simulations and a non-perturbative theory in H3H_3H3​, where the fourth order is regularized by hand, giving a complex energy to the virtual phonons of the four-phonon collisional processes. For a weakly interacting fluid and a phonon mode in the long-wavelength limit, we predict a damping rate about three times lower than that of the golden rule.
We study electron-phonon interaction induced decoherence between two-electron singlet and triplet states in a semiconductor double quantum dot using a spin-boson model. We investigate the onset and time evolution of this dephasing, and study its dependence on quantum dot parameters such as dot size and double dot separations, as well as the host materials (GaAs and Si). At the short time limit, electron-phonon interaction only causes an incomplete initial Gaussian decay of the off-diagonal density matrix element in the singlet-triplet Hilbert space; a complete long-time exponential decay due to phonon relaxation would eventually dominate over two-spin decoherence. We analyze two-spin decoherence in both symmetric and biased double quantum dots, identifying their difference in electron-phonon coupling and the relevant consequences.